數(shù)值分析-期末1.課件第4章

特插值/*HermiteInterpolation即：要求插值函數(shù)滿足xifxixifxi(x)f(x)(m)(x)

f(m)(x

i , 注：N

N10要求在1x0處直到m0階導(dǎo)數(shù)都重合的插0x

f

x0!

xx)m0

R(x)

f(x)(x)

f(m01)(

(x

)(m00(m0

x與

(x)的值

f

n函數(shù)

f(

i i以及導(dǎo)數(shù)值fn構(gòu)造不超過2n+1 i

(x) i0,滿足如下的2n+2個,

H2n1(xi)

f(xi

i0,1,H

(x)

f(x 2n1 思想Hermite思想設(shè)滿足如下2n+2個條件的插值多項式H2n1 H2n1(x)

ii0i

f(

)i(x)

i0

)i(x)其中i(

(xj

(xj)

i(

)

0

i, ,

(xj

i,

i(x和i(xi(x和i(x)均為2n+1次多項式，且有n個二重

,xAxB)l2x) 其中

(x)

(x

x0)(x

(x

xi1)(x

xi1 (x

xn (

x)(xx (x

)(x (xx

i

i (x)Al2(x)(AxB)l2(x)

l2(x)2l(x)l(x)i(xi)

Ai

Bi

(x)A2(AxB)l(

) 解之得

2xnnB B

k01ki1Ai

1

nn2xikki

1xi (xx)(x

(x

)(x

(xx l(x)

i

i (

x)(xx (x

)(x

(

x

i

i

ll(x)iik0kin1xikk0k

x

x

x0 (x

xi1)(x

xi1 (x

xn從而得到插值基函數(shù)ix)n(x)[12(xx) n x k0 kii0,1, ,

x)令xCxx)l2x) (x)Cl2(x)C(xx)l2(x) i i i xi,Ci,

(x)(xx)l2(x)j(xj(x)i(xj)ii,j,全導(dǎo)數(shù)的Hermite插值多項 H2n1(x)nn

i0

f(

)i(x)

i0

f(

)i(x)(x)[12(xx) ]l2(x) x k0 ki(x)(xx)l2(

i0,1, , 如n=1時Hermite插值多項式H3(x) xx

xx

xx

xx2H(x)f(x)12 1

f(x)12 0 x x

x x 1

1 0 0xx2 xx2f(

)(x

) 1

f(

)(x

) 0 x x 1 0H9(x)H9(x)yyf(x)y (xj)(x)i,j0,1,ijQuizxii+1,i012.2x)yy1 1斜率0.50123456x-0123456xH2n1)一 H2n1x也是滿足插值條件（*）的不超過2n+1x

H2n1(x)

H2n1(x)(xi)

i0,1, , i (x)i

xn+1 x

i0

值函 f(

[a,b]上有2n+1階連續(xù)

(2n2)(x

(x)fx互異節(jié)

xn[ i（*）的不超過2n+1次的插值多 i

x[a,

x[a

f(2n2)( (x)f(x)

(x)

2(x)2n1 2n1

(2n

2)!

證明方法同證明方法同例1：已知函數(shù)y

在點

012f(xi123f(xi-012 應(yīng)用Hermite插值計算f2

(x)

f(x

(x)f(x)(x)25i025

i0 (x)

x)

]l2(x) x k0 ki(x)(xx)l2(

i0,1, (

)][(

01 0 (01)(014

3x)(

1)2(

(

)][(

0)(

1x2(

1 1

0)(1(

)][(

2

21

(2

0)(21(73x)x2(4

(x)(x0)l2(x)1x(

1)2(

(x)(x1)l2(x)

(x1)x2(

(x)(x2)l2(x)1(

2)x2(

H5(x)

i0

f(

)i(x)

i0

f(

)i(x)1x25x25x31x41x5442

H5

例2：已知函數(shù)y

fx

xi(i

f(xif(x0f(x2f(xif(x0xH3(xi)

f(xi

i0,1,H(x)f(x 并推導(dǎo)其插值余項（已知f有4階連續(xù)導(dǎo)數(shù)） 首先構(gòu)造滿足插值條件Hx

f(

N2(x)

f(x0)

f[x0,x1](x

x0)

f[x0,x1,x2](x

x0)(x

令H3x

N2(x)

k(x

x0)(x

x2由H3x0

H3(xi)

f(

k(

k(x0

f(x0)

f[x0,x1]

f[x0

x2](

(x0

x2 R3(x)f(x)

H3(x)k(x)(xx)2(xx)(xx

g(t)

f(t)

(t)k(x)(tx)2(tx)(txg(t在至g(t在至少有4個互異零g(t

x(i=0,1 g(x0) (x

xi

g(4)(t)

至少有1 (4) (4)

f(4)( (t) (t)f(4)(

4!k(

k(x)4!R(x)

(x

)2(xx)(xx 4!

§4分段插值/*piecewiseInterpolation

f(n1)(Rn1(x)f(x)

Hn1(x)

(n

n1(x)插插值余項與節(jié)點的分布有關(guān)f足夠階連續(xù)導(dǎo)數(shù)（夠光滑），隨著節(jié)點個數(shù)的增加f(n1)例例3：在[5,5]fx1的L(x)。取x510i(i01ninL(x)fn210-4-3-2- n越大，端點附近變化越大，稱為Rungeynnn,,fxfx

(i

n由n

i0

f(

i

Lx)和Lx)n L(x)

L(x)f(x)l(x)f(x)l(x)

l(x) i

i0

i0取Newton kNn(x)

f(x0)

k

f[x0

xk](xj0

xj kN(x)f(x)f[x ,x](xx k

j0 N(x)N(x)[f(x)f(x k0{0k

f[x0

xk]

f[x

xk]}(xj0

xjy

y

Ck(k

k

xy

x0

kn n

f[

,x ,

]k

k!hk 0k

2

3 4000

k

0 000將插值區(qū)間劃分為若干個小區(qū)間（通常取等距劃分采用低次 x采用低次在區(qū)間[ab]上得到

i(x)f(x)

hmaxhmax(xi1xi

(x)

p0(x) p1(x)pn1(x)

xx0xx1xxn1

x1x2xnff(x)Pi(x)xiixii一、分段線性插值/*piecewiselinearinterpolation在每個區(qū)間[xixi1上，用1階多項式(直線fx[xi,xix[xi,xix[a,h(x)f(xj)lj(x)j0n分段l(x)jiiiPi(x)li(x)yili1(x)x

xxl(x)x 00

xx

0 xxx l0(xl0(x)y0l1(x)y1l2(x)xl(x)

xx x 0

x2

xxx

xjx

xj1

xx jlj(x)

xx

xjx

xj

xxj

j,n,n1

{a,b , } j

j1l(x)

xx

xn1

x 00

x0

x

ax0

x1

x2 xn

y,y,y ,

,fx)C1[a, f(x[a,b]

x[a,b]

(xi

yi

|R(x)

f(